Turán type inequalities for the partial sums of the generating functions of Bernoulli and Euler numbers

Abstract

Turán type inequalities for the partial sums of the generating functions of the Bernoulli and Euler numbers are established. They are shown to follow from a general result relating Turán inequalities of partial sums with Turán inequalities of the corresponding remainders in any Maclaurin expansion. Remainders in asymptotic expansions of the β‐function are shown to be completely monotonic of positive order.

On Completely Monotonic and Related Functions

“…This has been proven [51] by complex analysis methods. From (42) and (40), we easily derive an analogue of Proposition 16: An asymptotic expansion similar to (37) holds for the function β(x) itself.…”

“…. In order to demonstrate the usefulness of Theorem 13, let us consider an application obtained in [51].…”

“…An explicit expression for the functions ω n (t) has been found in [51] using the Cauchy residue theorem. Here, we can derive the same formula in a different way as follows: We observe that…”

Hypergeometric Euler numbers

“…as an analogue of (1). In [18], they are called weighted Bernoulli numbers, but this naming means different in other literatures. Since…”

“…When N = 0, then E n = E 0,n are classical Euler numbers defined in (1). In [18], the truncated Euler polynomial E m,n (x) is introduced as a generalization of the classical Euler polynomial E n (x). The concept is similar but without hypergeometric functions.…”